Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(4,8)(11,15)(13,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(33,34,37,38)(35,41,42,39)(36,40,44,46)(43,48,45,47);(2,6)(3,7)(9,10,12,14)(11,13,15,16)(18,22)(20,29)(23,25)(26,32)(34,38)(35,42)(36,44)(47,48);(1,2,5,6)(3,4,7,8)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,37,42)(34,39,38,41)(36,45,44,43)(40,48,46,47);(1,3,5,7)(2,4,6,8)(9,11,12,15)(10,13,14,16)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,36)(34,40)(35,43)(37,44)(38,46)(39,47)(41,48)(42,45);(1,4)(2,7)(3,6)(5,8)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,46)(45,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=i,c^2=g,d^2=fgi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4419605);